Yep. Woke is just a shorthand term for the identity politics that came out of Critical Theory. On the left it turned into DEI, BLM and Queer Theory. On the right it turned into identitarianism.
That said the problem with both the right and the left is that with the culture wars they've both just spiraled into post-truth tribalistic camps and radicalized themselves. The left especially became highly totalitarian and "liberal" or "progressive" are no longer suitable adjectives for them, although I'm sure they "identify" as being progressive they've regressed into worse sexist and racist stereotypes than even existed in the 1950s.